On the positiveness of a convolution product

Question

I've the following question:

Assume that $f(t),g(t)\in L^2[0;+\infty)$ and $f(t)$ is causal and positive $\forall t > 0$. Consider the convolution operation

$$\int_0^{+\infty} f(\tau)g(t-\tau) d\tau = y(t)$$

Is there any "compact" way to state the necessary conditions on $g(t)$, such as the convolution product $y(t)$ is positive $\forall t > 0$?

Thanks!